Chain rule with exponential and trigonometric functions

Let $f(x) = e^{\cos(x)}$ . Find $f'(x)$ .

Hint

Treat $f(x)$ as a composition $f(x) = e^{u}$ where $u = \cos(x)$ . Differentiate the outer exponential function first, leaving the inner function unchanged, then multiply by the derivative of the inner function.

Step-by-step solution

Identify the outer function as $e^{u}$ and the inner function as $u = \cos(x)$ . Recall that $\frac{d}{du}[e^u] = e^u$ and $\frac{d}{dx}[\cos(x)] = -\sin(x)$ .

By the chain rule, $f'(x) = \frac{d}{du}[e^u] \cdot \frac{du}{dx} = e^{\cos(x)} \cdot (-\sin(x))$ . Simplifying gives $f'(x) = -\sin(x) e^{\cos(x)}$ .

Chain rule with exponential and trigonometric functions

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